Influence of the regularity of the test functions for weak convergence in numerical discretization of SPDEs

This article investigates the role of the regularity of the test function when considering the weak error for standard discretizations of SPDEs of the form $dX(t)=AX(t)dt+F(X(t))dt+dW(t)$, driven by space-time white noise. In previous results, test functions are assumed (at least) of class $\mathcal{C}^2$ with bounded derivatives, and the weak order is twice the strong order. We prove, in the case $F=0$, that to quantify the speed of convergence, it is crucial to control some derivatives of the test functions, even when the noise is non-degenerate. First, the supremum of the weak error over all bounded continuous functions, which are bounded by $1$, does not converge to $0$ as the discretization parameter vanishes. Second, when considering bounded Lipschitz test functions, the weak order of convergence is divided by $2$, i.e. it is not better than the strong order. This is in contrast with the finite dimensional case, where the Euler-Maruyama discretization of elliptic SDEs $dY(t)=f(Y(t))dt+dB_t$ has weak order of convergence $1$ even for bounded continuous functions.